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{{redirect|Domain (mathematics)|other uses of "domain"|Domain (disambiguation)#Mathematics{{!}}Domain (disambiguation)}}
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[[Image:Codomain2.SVG|right|thumb|280px|Illustration showing ''f'', a function from pink domain ''X'' to blue and yellow codomain ''Y''. The smaller yellow oval inside ''Y'' is the [[Image (mathematics)|image]] of ''f''. Either the image or the codomain also sometimes is called the [[range (mathematics)|range]] of ''f''.]]
 
In [[mathematics]], the '''domain of definition''' or simply the '''domain''' of a [[function (mathematics)|function]] is the set of "input" or [[Argument of a function|argument]] values for which the function is defined.  That is, the function provides an "output"  or [[Value (mathematics)|value]] for each member of the domain.<ref>Paley, H. ''Abstract Algebra'', Holt, Rinehart and Winston, 1966 (p. 16).</ref>  The set of values the function may take is termed the [[Range (mathematics)|range]] of the function.  
 
For instance, the domain of [[cosine]] is the set of all [[real numbers]], while the domain of the [[square root]] consists only of numbers greater than or equal to 0 (ignoring [[complex numbers]] in both cases). For a function whose domain is a subset of the [[real numbers]], when the function is represented in an ''xy'' [[Cartesian coordinate system]], the domain is represented on the ''x''-axis.
<!-- The image and caption below are problematic - see the discussion -->
<!--[[Image:Square_root_0_25.svg|thumb|250px|The function ''f''(''x'') = &radic;<span style = "text-decoration:overline">''x''</span> has domain that consists of all real numbers between 0 and positive infinity]]-->
 
== Formal definition ==<!-- This section is linked from [[Complex analysis]] -->
Given a [[Function (mathematics)|function]] ''f'':''X''→''Y'', the set ''X'' is the '''domain''' of ''f''; the set ''Y'' is the [[codomain]] of ''f''. In the expression ''f''(''x''), ''x'' is the '''argument''' and ''f''(''x'') is the '''value'''. One can think of an argument as an input to the function, and  the value as the output.
 
The [[image (mathematics)|image]] (sometimes called the [[range (mathematics)|range]]) of ''f'' is the set of all values assumed by ''f'' for all possible ''x''; this is the set {{nowrap begin}}{''f''(''x'') | ''x'' &isin; ''X''}{{nowrap end}}.  The image of ''f'' can be the same set as the codomain or it can be a proper subset of it.  It is in general smaller than the codomain; it is the whole codomain if and only if ''f'' is a [[surjective function]]. 
 
A well-defined function must map every element of its domain to an element of its codomain. For example, the function ''f'' defined by
: <math>f(x)=1/x</math>
has no value for ''f''(0).
Thus, the set of all [[real number]]s, '''R''', cannot be its domain.
In cases like this, the function is either defined on '''R'''&minus;{0} or the "gap is plugged" by explicitly defining ''f''(0).
If we extend the definition of ''f'' to
:<math>f(x) = \begin{cases}
1/x&x\not=0\\
0&x=0
\end{cases}</math>
then ''f'' is defined for all real numbers, and its domain is <math>\mathbb{R}</math>.
 
Any function can be restricted to a [[subset]] of its domain.
The [[Restriction (mathematics)|restriction]] of ''g''&nbsp;:&nbsp;''A''&nbsp;→&nbsp;''B'' to ''S'', where ''S'' ⊆ ''A'', is written ''g''&nbsp;|<sub>''S''</sub>&nbsp;:&nbsp;''S'' →&nbsp;''B''.
 
==Natural domain==
The '''natural domain''' of a formula is the set of values for which it is defined, typically within the reals but sometimes among the integers or complex numbers. For instance the natural domain of square root is the non-negative reals when considered as a real number function. When considering a natural domain the set of possible values of the function is typically called its range.<ref>{{cite book|title=Calculus: basic concepts and applications|first1=Robert A.|last1=Rosenbaum|first2=G. Philip|last2=Johnson|page=60|year=1984|isbn=0-521-25012-9|publisher=Cambridge University Pressd}}</ref>
 
==Domain of a partial function==
{{further2|[[Partial function#Domain of a partial function]]}}
There are two distinct meanings in current mathematical usage for the notion of the domain of a [[partial function]] from ''X'' to ''Y'', i.e. a function from a subset ''X''<nowiki>'</nowiki> of ''X'' to ''Y''. Most mathematicians, including [[recursion theory|recursion theorists]], use the term "domain of ''f''" for the set ''X''<nowiki>'</nowiki> of all values ''x'' such that ''f''(''x'') is defined. But some, particularly [[category theory|category theorists]], consider the domain to be ''X'', irrespective of whether ''f''(''x'') exists for every ''x'' in&nbsp;''X''.
 
== Category theory ==
 
In [[category theory]] one deals with [[morphisms]] instead of functions. Morphisms are arrows from one object to another.  The domain of any morphism is the object from which an arrow starts.  In this context, many set theoretic ideas about domains must be abandoned or at least formulated more abstractly.  For example, the notion of restricting a morphism to a subset of its domain must be modified.  See [[subobject]] for more.
 
== Real and complex analysis ==
 
In [[calculus|real]] and [[complex analysis]], a domain is an [[open set|open]] [[connected space|connected]] subset of a [[real number|real]] or [[complex number|complex]] vector space.
 
In [[partial differential equation]]s, a domain is an open connected subset of the [[euclidean space]] R<sup>n</sup>, where the problem is posed, i.e., where the unknown function(s) are defined.
 
== More examples ==
* Function <math>f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = \sqrt{x}</math> is defined for all <math>x \geq 0</math>.
* Function <math>f: \mathbb{R} \rightarrow \mathbb{C}, f(x) = \sqrt{x}</math>, where <math>\mathbb{C}</math> is the set of all [[complex numbers]], is defined for all ''x''.
* Function <math>\tan x = \frac{\sin x}{\cos x}</math> is defined for all <math>x \neq \frac{\pi}{2} + k \pi, k = 0, \pm 1, \pm 2, \ldots</math>
 
==See also==
 
* [[Range (mathematics)]]
* [[Codomain]]
* [[Surjective function]]
* [[Injective function]]
* [[Bijection]]
* [[Domain decomposition]]
* [[Lipschitz domain]]
* [[Effective domain]]
 
==References==
<references/>
 
{{logic}}
 
[[Category:Functions and mappings]]
[[Category:Basic concepts in set theory]]

Latest revision as of 00:16, 24 August 2014

Using jewelry and training the skill of adornment is definitely an age group-old process that has roots in social, social, and design motions and history. Regardless of whether you are looking for insight on the way to wear jewelry to improve your individual design, how to purchase the very best jewels, or maybe for interesting facts, you may unquestionably, find some intriguing and helpful gems over these hand-picked suggestions.

Pewter precious jewelry has become broadly well-known for years. If you're seeking to invest in expensive jewelry pieces created from pewter, make sure that they are guide free of charge. For a long period, pewter jewellery has included steer triggering a lot of harmful side effects for the person wearing them. Typically contemporary precious jewelry made from this steel is promoted as guide-free of charge if it doesn't include any.

If you have aluminum allergy symptoms, ensure that you obtain nickel free expensive jewelry. Nickel is a kind of alloy that is combined together with treasured metals such as silver to ensure they are more flexible. Gold is really water in their original state and so not significant enough to become correctly shaped into precious jewelry. Often, one more alloy like nickel is included. Ensure you check out what you are acquiring to ensure it is nickel cost-free.

The world of body piercing has brought physique precious jewelry to new altitudes. Lots of people just like the sensation of having their tongues pierced, but getting aluminum expensive jewelry with your jaws might be harmful in your tooth. There are many colourful possibilities for plastic tongue jewelry that may be less dangerous for your mouth area and won't help make your dentist cringe.

When picking expensive jewelry to use for your wedding, make a decision initially in order to buy it or rent payments it. Most of the expensive jewelry you dress in as being a bride isn't actually anything you'd dress in yet again, so booking is a wonderful option and can get you a better grade of product than you might be able to acquire straight up.

Buy significant jewellery for almost any occasion. Here's more info in regards to Costume Jewelry take a look at our page. Providing jewelry as being a gift item results in an extensive enduring effect on the designed receiver. Nonetheless, you can always give a individual touch by having the jewellery part engraved using a personal meaning or add a elegance of great value. Personalized jewellery given being a gift idea constantly makes a supportive gesture.

To help expensive jewelry preserve its highest probable benefit you need to try to make it in a clean condition. Not only will it look greater to you but it will fetch a greater asking price in the event you actually plan to sell it. Semi twelve-monthly travels to jewelry shop for normal cleanings may help do this.

You may have a more wonderful diamonds when it is lower effectively, instead of a diamond of any greater dimensions. You need to also consider the persona and personal preferences from the gift beneficiary.

Become knowledgeable about the gems you are interested in well before will make a brand new purchase of jewelry. You have to know what questions you should ask in order to prevent being taken advantage of with a jeweler. Practically nothing seems much worse than being cheated for something you imagined was a superior quality product.

When deciding the sort of expensive jewelry you need to buy for your significant other, it is wise to keep in mind that good quality does matter. While you would be wise to attempt to help make your buy an affordable a single, you do not desire to sacrifice quality. In case you have a established selling price, research prices for a good price.

If your precious jewelry is causing skin discomfort, there are particular actions you can take to lower it. Use a tiny amount of powder on the skin in the area the precious jewelry is going to be touching. This assists absorb dampness, which can then lessen the probability of allergy symptoms or tenderness.

A lot of treasured gemstones and stones cannot be place in a washing answer. It is important that you recognize the structure of your own engagement ring as well as the formula of your respective washing remedy. Some stones like pearl, opal, turquoise and also other soft gemstones can get destroyed by the tough substances located in expensive jewelry more clean.

When attempting to consider new jewellery styles, just shop around you! You'll be blown away at how straightforward it is to find motivation from nature and your surroundings. Take photographs of items that encourage you to definitely make the expensive jewelry to be able to guide it with ease later on. You will find inspiration for a variety of things, including colour palettes and forms.

Teaching yourself will enable you to get the best of your own precious jewelry activities. It is possible to go forwards from this point content in knowing you are more intelligent than you were a couple of minutes in the past. Remember these helpful expensive jewelry ideas, and you will definitely find you might have some terrific precious jewelry at the end.